Intermediate Macroeconomics Problem Set

1. Consider the following two period problems where consumers have preferences over consumption and leisure:

max_{c0,c1,a1,a2,l0,l1} lnc₀ + ln(z^- - l₀) + β[ln c₁ + ln(z^- - l₁)]

s.t. c₀ + a₁ ≤ (1 + R)a₀ + (1 - τ₀)w₀l₀

c₁ + a₂ ≤ (1 + R)a₁ + (1 - τ₁)w₁l₁

a₂ ≥ 0, a₀ = a¯₀

i) Construct the Lagrangian for this problem.

ii) Take the first order conditions and find the Euler Equation for labor and the Euler Equation for consumption. (These are intertemporal conditions obtained from the FOCs for consumption and labor.)

iii) Using the Euler equation for labor, argue that if τ₁ goes up, then the consumer works less in period 1.

2. Consider a T period problem where consumers have preferences over consumption and leisure. Consumers can work in competitive firms which hires labor and capital. There is no government sector.

i) Define a sequential market equilibrium for this economy.

ii) Define a Arrow Debreu competitive equilibrium for this economy.

iii) Derive the Euler equation for consumption. Explain how does this show that the cost and benefit of savings in equalized by the agent.

iv) Derive the intratemporal condition between labor and consumption.

3. Consider a T period problem where consumers have preferences over consumption and leisure. Consumers can work in competitive firms which hires labor and capital. The government needs to finance a stream of government expenditure {g_t}^T_t=0 = {g₀, g₁, ....g_T}.

In Economy 1, the government uses only consumption taxes at the rate {τ_ct}^T_t=0

In Economy 2, the government only taxes capital income at the rate {τ_kt}^T_t=0

i) For Economy 1, define the Arrow Debreu competitive equilibrium.

ii) For Economy 1, set up the Lagrangian, take first order conditions and find the Euler equation for consumption.

iii) For Economy 2, define the Arrow Debreu competitive equilibrium.

iv) For Economy 2, set up the Lagrangian, take first order conditions and find the Euler equation for consumption.

v) Find a relationship between the tax rates in the two economies such that the allocations are identical. (This involves finding a condition such that the Euler Equation for the two economies are identical.)

4. Consider a T period problem where consumers have preferences over only consumption. Labor is inelastically supplied, 't = 1 for all t. Consumers can work in competitive firms which hires labor and capital. The government needs to finance a stream of government expenditure {g_t}^T_t=0 = {g₀, g₁, ....g_T}. Assume that the government uses a constant tax, τ, applied to all (i.e., both labor and capital) income uniformly. Assume that the proceeds of this tax are used, balancing the budget in every period, to purchase the stream of purchases of goods, g_t.

i) Define the Arrow Debreu competitive equilibrium for this economy.

ii) Write down the first order conditions for the household problem, firm problem and government budget balance.

iii) Derive the Euler equation for consumption.

iv) Characterize the steady state of this equilibrium.

[Hint: Steady state refers to a situation where the variables are not changing over time. The competitive equilibrium is characterized by the Euler equation for consumption, the 2 firm optimality condition, the government budget balance condition and the market clearing condition for the final good. So write down these conditions without the t subscripts and simplify.]